group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
Nonabelian groupoid cohomology is the nonabelian cohomology of groupoids.
This generalizes nonabelian group cohomology, which is the special case of nonabelian group cohomology for groupoids that are deloopings $\mathbf{B}G$ of groups $G$.
The Atiyah Lie groupoid $At(P) \to \Pi(X)$ of a principal bundle $P \to X$ is the extension of $\Pi(X)$ classified by the nonabelian groupoid cocycle $\Pi(X) \to \mathbf{B} AUT(G)$ coming from a connection on $P$.